AVERYSIMPLEAPPROACHFOR3-DTO2-DMAPPING

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1 Image Processing& Communications, vol. 11, no. 2, pp AVERYSIMPLEAPPROACHFOR3-DTO2-DMAPPING SANDIPANDEY (1),AJITHABRAHAM (2),SUGATASANYAL (3) (1) AnshinSoftwarePvt.Ltd. INFINITY, Tower- II, 10th Floor, Plot No Block- GP, Salt Lake Electronics Complex, Sector-V,Kolkata (2) IITAProfessorshipProgram,SchoolofComputerScience, Yonsei University, 134 Shinchon-dong, Sudaemoon-ku, Seoul , Republic of Korea (3) SchoolofTechnology&ComputerScience Tata Institute of Fundamental Research Homi Bhabha Road, Mumbai , INDIA Abstract. Manytimesweneedtoplot3-Dfunctionse.g.,in many scientific experiments. To plot this 3-D functionson2-dscreenitrequiressomekindofmapping. Though OpenGL, DirectX etc 3-D rendering libraries have made this job very simple, still these libraries come with many complex pre-operations that are simply not intended, also to integrate these librarieswithanykindofsystemisoftenatough trial. This article presents a very simple method of mappingfrom3-dto2-d,thatisfreefromanycomplex pre-operation, also it will work with any graphics system where we have some primitive 2-D graphics function. Also we discuss the inverse transform and how to do basic computer graphics transformations using our coordinate mapping system.

2 76 S. Dey, A. Abraham, S. Sanyal 1 Introduction 2 Proposed approach We have a pictorial representation(fig.1) of our 3-D to 2-D mapping system: Wehaveafunction f :R 2 R, andourintention istodrawthefunctionin2-dplane. Thefunction z = f(x,y)isa2-variablefunctionandeachtuple (x,y,f(x,y)) R 3.Let ssaywewanttographically plot f onto computer screen using a primitive graphics library (like Turbo C graphics), which supports only the basic putpixel (to draw a pixel in 2-D screen)-like 2-D rendering function, but no 3-D rendering; i.e., our graphics library s putpixel s domainisr 2 andit snotr 3. Fig.1:BasicModelofasimple3-Dto2-Dmapping system But, how the function f should look like after mapping and plotting? Here we simulate the 3-rd coordinate(namelyz)inour2-dx yplane. We perform the logical to physical coordinate transform andeverythingbythemapfunctionh, whichwill basicallyturnouttobea3 2matrix. Thebasic Henceinordertodrawthefunction f usingour graphics library, we must design a coordinate conversion system, that will provide us with a functionthatwilltakeasinput3-tuples (x,y,f(x,y))and produceasoutputa2-tuple (x,y )thatcanbedirectlypassedtoourgraphicslibrarytoplotitonto thescreen, butwith3-dlook&feel. Aswediscussed, it s essential that we have a simple coordinatemappingsystemthatmapsr 3 tor 2 andstill mappingtechniqueisshowninfig.2,whichweare shortly going to explain. IfwehaveourOrigin0at (x 0,y 0 )screencoordinate, we have, x =x 0 +y x sin(θ) (1) y =y 0 z+x cos(θ) i.e.,wehaveour3-dto2-dtransformationmatrix: gives us a hypothetical feeling of drawing 3-D functions.it sveryeasytofindsuchamap,i.e.,afunctionh:r 3 R 2 andinthispaperwetrytofindsuch a simple map. M 3 2 = sin(θ) cos(θ) (2)

3 Image Processing& Communications, vol. 11, no. 2, pp controlthedimensionalongz axisbyacompressionfactor ρ z andslightlymodifyingtheequations: x =x 0 +y x sin(θ) y =y 0 ρ z z+x cos(θ) (4) Obviously,0.0 < ρ z 1.0 Bydefaultwetake ρ z = Sample output surfaces drawn using the above mapping Followingsurfaces(Fig. 3andFig. 4)aredrawn in Turbo C++ version 3.0(BGI graphics) using the above simple 3-D to 2-D mapping. Fig. 2: The basic coordinate mapping Againwehaveshifting(changeoforigin)bythe matrixo 2D = x 0,y 0 sothato 2D +P 3D M 3 2 = P 2D,here denotesmatrixmultiplicationand +denotesmatrixaddition, the3-tuplep 3D = xyz, the 2-tupleP 2D = x y i.e. x 0 y 0 + x y z. sin(θ) cos(θ) = x y (3) By default we keep the angle between X axis & Z axis = θ = π 4, that one can change if required, but with the following inequality strictlysatisfied:0 < θ < π 2. 4 Inverse Transformation- Obtaining original 3-D coordinates from the transformed 2-D coordinates One can optionally use a compression factor to Here, our transformation function (matrix) is definedbyeqn.(1).

4 78 S. Dey, A. Abraham, S. Sanyal Fig.3:SinefunctiondrawninTurboC++Version3.0(BGIGraphics)usingthe3-Dto2-Dmapping Fig.4:SyncfunctiondrawninTurboC++Version3.0(BGIGraphics)usingthe3-Dto2-Dmapping

5 Image Processing& Communications, vol. 11, no. 2, pp Aswecansee,itisimpossibletore-convertandobtain the original set of coordinates, namely (x, y, z), becausewehave3unknownsand2equations. So, in order to be able to get the original coordinates back,weatleastneedtostore3tuplesasresultof thetransformation,forinstance, (x,y,z) (x,y,z), the z-coordinate being stored only to get the inverse transform (x,y,z) (x,y,z)andthe (x,y )pairis usedtoplotthepoint.so,inordertogettheinverse transformation, we need to solve the equations for x,y,sincewealreadyknowz,wehave2-equations and 2 unknown variables: sin(θ) cos(θ) 0 M 3 3 = with sin(θ) cos(θ) 0 Det(M 3 3 ) =det = (7) sin(θ) cos(θ) =1 = cos(θ) (8) 1 0 Now, 0 < θ < π 2, hence cos(θ) 0, hence Det(M 3 3 ) 0andtheinverseexists. y x sin(θ) =x x 0 x cos(θ) =y y 0 +z solving the above 2 equations we get, (5) x 0 y x y z. sin(θ) cos(θ) = = x y z (9) x = (y y 0 +z) sec(θ) (6) y =x x 0 +(y y 0 +z) tan(θ) Put it in another way, our transformation matrix isa3 2matrixandisdonebyEqn. (2)sincea non-square matrix, no question of existence of its inverse. So, inordertobeabletogettheinverse transformaswell,weneeda3 3invertiblesquare matrix, e.g., But, we have, Inv(M 3 3 ) = (M 3 3 ) 1 = Adj(M 3 3) Det(M 3 3 ) Det(M 3 3 ) 0 (10) and, 0 cos(θ) 0 Adj(M 3 3 ) = 1 sin(θ) 0 (11) 1 sin(θ) cos(θ)

6 80 S. Dey, A. Abraham, S. Sanyal Hence, Inv(M 3 3 ) = (M 3 3 ) 1 = sec(θ) tan(θ) 0 sec(θ) tan(θ) 1 (12) 5 Rotation and affine transformations Apointin3-D,afterbeingmappedto2-Dscreen, following the above mapping procedure, may be required to be transformed using standard computer graphics transformations(translation, rotation here,cos(θ) 0. So, the inverse transform is: aboutanaxisetc). Butinordertoundergosucha graphics transformation and to show the point back to the screen after the transformation, it needs to sin(θ) cos(θ) 0 x y z = = x y z x 0 y 0 0 (13) go through the following steps in our previouslydescribed coordinate mapping system: First obtain the inverse coordinate transformation to obtain the original 3-D coordinates from the mapped 2-D coordinates. Multiply the 3-D coordinate matrix by proper = x y z = x x 0 y y 0 z sec(θ) tan(θ) 0 sec(θ) tan(θ) 1 (14) graphics transformation matrix in order to achieve graphical transformation. Usethesame3-Dto2-Dmapagaintoplotthe point onto the screen. x y z = (y y 0 +z) sec(θ) x x 0 +(y y 0 +z) tan(θ) z (15) This exactly matches with our previous derivation.

7 Image Processing& Communications, vol. 11, no. 2, pp These steps can be mathematically represented as: P 3D =P 2D (M 3 3 ) 1 P 3D =P 3D T 3 3 P 2D =P 3D M 3 3 Or, by a single line expression, P 2D = ((P 2D (M 3 3 ) 1 ) T 3 3 ) M 3 3 ) Here, as before denotes matrix multiplication, where T 3 3 denotes the traditional graphics transformation matrix. But, since we know the fact that matrix multiplication is associative, we have, P 2D = ((P 2D (M 3 3 ) 1 ) T 3 3 ) M 3 3 =P 2D (M 3 3 ) 1 T 3 3 M 3 3 (16) =P 2D (M 3 3 ) 1 T 3 3 M 3 3 ) P 2D =P 2D M wherem = (M 3 3 ) 1 T 3 3 M 3 3 ThismatrixM isneededtobecomputedoncefor a given graphics transformation(e.g., rotation about anaxis)andappliedtoallpointsonthescreen,so that using a single matrix multiplication thereafter any point on the screen can undergo graphics transformation,by,p 2D =P 2D M,whereP 2D represents the point mapped before transformation T 3 3 and P 2D isthepointre-mappedafterthetransformation, as obvious. Hence,usingtheabovetricksweareabletomake the transformation more computationally efficient. Moreover, if a transformation is needed to be applied simultaneously, we can use the property M (T 3 3) n M 3 3 = (M T 3 3 M 3 3 ) n, where (T 3 3 ) n denotes(ntimes,nisapositiveinteger)simultaneousmatrixmultiplicationoft 3 3. Let ssaywehavealreadyundergonet 3 3 atransformation,sothatwehavealreadycomputedm = (M T 3 3 M 3 3,andlet ssaythatwealsohave frequent simultaneous (T 3 3 ) n transformation. In ordertoundergoa(t 3 3 ) n transformation,wefirst So, using this simple technique we can escape the 3 successive matrix multiplications every-time a point on screen needs to transformed - instead we can pre-compute the matrix M = (M 3 3 ) 1 T 3 3 M 3 3. needtocomputethematrix (T 3 3 ) n,thenweneed tocomputeournewmatrixm = (M (T 3 3) n M 3 3,soweneedtotaln+2matrixmultiplications, every-timewewanta(t 3 3 ) n transform,foreachn. ButifwehavecomputedM T 3 3 M 3 3 initially, herethetrickisthatwecanreusethisitto

8 82 S. Dey, A. Abraham, S. Sanyal compute our new matrix in the following manner: M =M (T 3 3) n M 3 3 = = (M T 3 3 M 3 3 ) n = (M ) n Hereweneednotcompute (T 3 3 ) n andm everytime,insteadweneedtocompute (M ) n only(that can be incremental multiplication to increase efficiency). 6 Conclusions This article presented a very simple method of mappingfrom3-dto2-d,thatisfreefromanycomplex pre-operation. The proposed technique works with any graphics system where we have some primitive 2-D graphics function. We also discussed the inverse Tom Davis, OpenGL Programming Guide, The Official Guide to Learning OpenGL, Version 1.4, Fourth Edition. 4 KenTurkowski,TheUseofCoordinateFrames in Computer Graphics, Graphics Gems I, Academic Press, 1990, pp Ken Turkowski, Fixed-Point Trigonometry with CORDIC Iterations, Graphics Gems I, Academic Press, 1990, pp C.M.Ng,D.W.Bustard,ANewRealTimeGeometric Transformation Matrix and its Efficient VLSI Implementation, Computer Graphics Forum,Volume13Page285 transform and how to do basic computer graphics transformations using our coordinate mapping system. 7 References 1 David F. Rogers, J. Alan Adams, Mathematical Elements for Computer Graphics, McGraw- Hill 2 David F. Rogers, Procedural elements for computer Graphics, United States Naval Academy, Annapolis, MD 3 Dave Shreiner, Mason Woo, Jackie Neider,

A Very Simple Approach for 3-D to 2-D Mapping

A Very Simple Approach for 3-D to 2-D Mapping A Very Simple Approach for -D to -D appig Sadipa Dey (1 Ajith Abraham ( Sugata Sayal ( Sadipa Dey (1 Ashi Software Private Limited INFINITY Tower II 10 th Floor Plot No. - 4. Block GP Salt Lake Electroics